Let this be a good lesson in the difference between semantics and syntax:
One would never say such a thing about ones teeth. It is a very unnatural suggestion; if you are in a strange enough position for others to be brushing your teeth (you have an illness or have had an accident that prevents you from brushing your teeth yourself, allowing others to brush your teeth is probably not something you can choose to do (the actual 'letting') - presumably others will decide that you need to have your teeth brushed.
But...the general syntax is just fine. "Let (something) (present subjunctive)" is a very natural structure:
- Let the boy tie his own shoelaces (don't do it yourself)
- Let them eat cake (it's your birthday, or you are sarcastically insulting poor people)
- Let it be (zen-like advice to others while your rock band falls apart)
That's how you use the imperative 'Let'. It does not sound old-fashioned at all.
In the particular sentence you gave it sounds strange because the semantics is strange, not the syntax.